Equilibrium Thermal Fluctuations Research Articles

A finite element framework for computation of protein normal modes and mechanical response

A computational framework based on the Finite Element Method is presented to calculate the normal modes and mechanical response of proteins and their supramolecular assemblies. Motivated by elastic network models, proteins are treated as continuum elastic solids with molecular volume defined by their solvent-excluded surface. The discretized Finite Element representation is obtained using a surface simplification algorithm that facilitates the generation of models of arbitrary prescribed spatial resolution. The procedure is applied to a mutant of T4 phage lysozyme, G-actin, syntenin, cytochrome-c', beta-tubulin, and the supramolecular assembly filamentous actin (F-actin). Equilibrium thermal fluctuations of alpha-carbon atoms and their inter-residue correlations compare favorably with all-atom-based results, the Rotational-Translational Block procedure, and experiment. Additionally, the free vibration and compressive buckling responses of F-actin are in quantitative agreement with experiment. The proposed methodology is applicable to any protein or protein assembly and facilitates the incorporation of specific atomic-level interactions, including aqueous-electrolyte-mediated electrostatic effects and solvent damping. The procedure is equally applicable to proteins with known atomic coordinates as it is to electron density maps of proteins, protein complexes, and supramolecular assemblies of unknown atomic structure.

Absolute Negative Mobility Induced by Thermal Equilibrium Fluctuations

A novel transport phenomenon is identified that is induced by inertial Brownian particles which move in simple one-dimensional, symmetric periodic potentials under the influence of both a time periodic and a constant, biasing driving force. Within tailored parameter regimes, thermal equilibrium fluctuations induce the phenomenon of absolute negative mobility (ANM), which means that the particle noisily moves backwards against a small constant bias. When no thermal fluctuations act, the transport vanishes identically in these tailored regimes. ANM can also occur in the absence of fluctuations on grounds which are rooted solely in the complex, inertial deterministic dynamics. The experimental verification of this new transport scheme is elucidated for the archetype symmetric physical system: a convenient setup consisting of a resistively and capacitively shunted Josephson junction device.

Anisotropy in Sickle Hemoglobin Fibers from Variations in Bending and Twist

We have studied the variations of twist and bend in sickle hemoglobin fibers. We find that these variations are consistent with an origin in equilibrium thermal fluctuations, which allows us to estimate the bending and torsional rigidities and effective corresponding material moduli. We measure bending by electron microscopy of frozen hydrated fibers and find that the bending persistence length, a measure of the length of fiber required before it starts to be significantly bent due to thermal fluctuations, is 130microm, somewhat shorter than that previously reported using light microscopy. The torsional persistence length, obtained by re-analysis of previously published experiments, is found to be only 2.5microm. Strikingly this means that the corresponding torsional rigidity of the fibers is only 6x10(-27)Jm, much less than their bending rigidity of 5x10(-25)Jm. For (normal) isotropic materials, one would instead expect these to be similar. Thus, we present the first quantitative evidence of a very significant material anisotropy in sickle hemoglobin fibers, as might arise from the difference between axial and lateral contacts within the fiber. We suggest that the relative softness of the fiber with respect to twist deformation contributes to the metastability of HbS fibers: HbS double strands are twisted in the fiber but not in the equilibrium crystalline state. Our measurements inform a theoretical model of the thermodynamic stability of fibers that takes account of both bending and extension/compression of hemoglobin (double) strands within the fiber.

Modeling context-dependent conformation parameters of DNA duplexes

The context-dependent conformation preference and the conformational lability of DNA play important roles in the formation of complexes with various proteins including transcription factors, regulatory and nucleosomal proteins. The extraction of context-dependent DNA helix parameters and the related deformation characteristics from static crystalline DNA duplex structures involves artifacts caused by the crystal packing effects. A series of 150 tetradeca-and pentadecameric DNA duplexes with various sequences occurring in aqueous solution under physiological conditions have been simulated by the method of molecular dynamics (MD). The context-dependent conformation parameters of XY dinucleotides were determined from analysis of the MD trajectories of equilibrium thermal fluctuations of 3D structures for a large set of 14-membered DNA duplexes in aqueous solution. The mean values of the helix parameters roll and twist as well as their thermal fluctuations for the TA, TG, and CG pair steps proved to take special values significantly differing from the averages over all pairs. This fact suggests that DNA sites rich in TA, TG, and CG are capable of forming special local equilibrium structures and have increased conformational plasticity, thus favoring conformational matching at the sites of regulatory and nucleosomal protein binding.

Transport and diffusion on crystalline surfaces under external forces

We present a numerical study of classical particles obeying a Langevin equation and moving on a solid crystalline surface under an external force that may either be constant or modulated by periodic oscillations. We focus on the particle drift velocity and diffusion. The roles of friction and equilibrium thermal fluctuations are studied for two nonlinear dynamical regimes corresponding to low and to high but finite friction. We identify a number of resonances and antiresonances, and provide phenomenological interpretations of the observed behaviour.

The critical point of an exothermic reactor

Conditions for quenching and igniting an exothermic reactor differ from one another, since a hysteresis takes place. In the parameter plane, the bistability region is represented as a peninsula terminating in a cusp. At the cusp point, the difference between hightemperature and low-temperature regimes disappears. In this respect, the cusp is similar [1] to the critical point of a phase transition (e.g., the liquid–vapor transition) in which the distinction between phases disappears. As is well known, density fluctuations are strong at the critical point. Similarly, we can expect an increase in the chaotic pulsations of temperature and concentration in the neighborhood of the cusp. In contrast to equilibrium thermal fluctuations in the case of phase transitions, these pulsations are caused by small random variations in the composition and other parameters of the incoming mixture. Usually, such variations do not lead to marked changes in the reactor state. However, near the critical point in which the reactor susceptibility is especially high, the pulsation intensity increases infinitely when approaching this point. In this study, we investigated the chaotic behavior of the reactor in the critical region of parameters. The similarity with the critical point of phase transition relates to reactor steady-state regimes. Under certain conditions, the region of periodic regimes [2–4] can include a cusp. Below, we also consider the effect of this factor.

Uniformity of the phase space and fluctuations in thermal equilibrium

General relations are found between the measure of the uniformity of distributions on the phase space and the first moments and correlations of extensive variables for systems close to thermal equilibrium. The role played by the parameter of the Renyi entropy for the analysis of their fluctuations and correlations is studied. Analytical results are verified and illustrated by direct simulations of quantum systems of ideal fermions and bosons. Problems of finite statistics, usual in experiments and simulations, are addressed and discussed and solved by finding unbiased estimators for Renyi entropies and uniformities.

Dynamics of step fluctuations on Pd( [formula omitted]) surface as observed by STM

The equilibrium thermal fluctuations of steps on the Pd(1 1 0) surface are studied by scanning tunneling microscopy (STM) at room temperature. A sequence of atomically resolved STM images of the Pd(1 1 0) surface was taken at a rate of one image every 370 s. Each image shows the same area of size 200 A ̊ ×200 Å and exhibits terraces and monatomic steps. On the Pd(1 1 0) surface at room temperature, individual Pd atoms at step edges can be visible in images. Steps do not move too much on the time scale required for a tunnel image. Nevertheless, it is revealed that step movement takes place from a comparison of the STM images that were taken in a time interval of few minutes. A statistical analysis of step dynamics on Pd(1 1 0) surface are performed.

Fluctuations in Self-Assembled Sickle Hemoglobin Fibers

We discuss microscopic measurements of thermal fluctuations in self-assembled sickle hemoglobin fibers. These permit the measurement of various mechanical moduli that control the rheology of the fiber gel responsible for the pathology of the disease. Differential interference contrast microscopy of isolated fibers undergoing equilibrium thermal bending fluctuations leads to estimates of thermal persistence lengths of between 0.23 and 10.4 mm for the most flexible and stiffest fibers, respectively. We argue that this large range may reflect the formation of fiber bundles. If the most flexible fibers are single fibers, then the stiffest measured object is consistent with a “closed shell” hexagonal bundle of 7 single fibers. We estimate the spectrum of persistence lengths associated with bundles consisting of different numbers of single fibers and compare this with the experimental data. The equivalent Young's modulus for the material is 0.1 GPa, less than for structural proteins but much larger than for ext...

Time-Asymmetric Fluctuations of Light and the Breakdown of Detailed Balance

Temporal fluctuations of the light radiated by a photoemissive source are studied through the cross correlation of output fields. Whereas microscopic reversibility guarantees time-symmetric fluctuations in thermal equilibrium--where detailed balance holds--away from equilibrium time asymmetry is permitted. Examples of time asymmetry in cavity QED are reported.

Experimental study of noise and Josephson oscillation linewidths in bicrystal YBCO junctions

The intensities of the noise in a bicrystal high-TC (HTS) Josephson junction have been precision-measured at 1–2 GHz frequency band at bias voltages up to 50 mV at T = 4.2 K. At large bias voltages, V > 30 mV, the dependence of current noise density was found exactly coinciding with the Schottky shot noise asymptote 2eI. At relatively low voltages, V < 4 mV, a noticeable noise rise has been registered. The broadening of Josephson oscillation linewidths ΔfJ over the values ΔfRSJ predicted by the RSJ model has been experimentally studied at different frequencies in the mm and submm wave range up to voltages V = 2 mV in connection with low-voltage noise rise. Both the features observed, the linewidth broadening and the excess noise over the noise level of thermal fluctuations, are discussed in terms of multiple Andreev reflection, giving rise to a nonequilibrium shot noise—the case which may take place in the d-wave superconducting junctions. Experimental results on noise performance are also compared with the qualitatively similar dependences of the current noise, known for the s-superconducting ballistic point-like or diffusive-type SNS junctions, where the excess low-voltage noise is manifested due to multiple Andreev reflections. Increasing the operating temperature, the thermal (equilibrium) fluctuations were found to predominate, resulting in a decrease of ratio ΔfJ/ΔfRSJ. The characteristics of the ac Josephson effect in HTS junctions measured at submm wave frequencies at temperatures close to the transition temperature TC are also discussed.

Dynamical properties of α-amylase in the folded and unfolded state: the role of thermal equilibrium fluctuations for conformational entropy and protein stabilisation

A comparative analysis of thermal equilibrium fluctuations occurring in a mesophilic and in a thermophilic α-amylase was performed to study the effect of structural fluctuations on thermostability. The thermal fluctuations determining the conformational entropy of both enzymes have been characterised for the folded (at 30°C and 60°C) and for the unfolded state by applying neutron spectroscopy (at 30°C). The folded state shows a higher structural flexibility for the thermophilic protein as compared to the mesophilic homologue. In contrast, the unfolded state of both enzymes is rather similar with respect to the structural fluctuations. On the basis of this result, a mechanism characterised by entropic stabilisation (i.e., smaller Δ S for the unfolding transition of thermophilic α-amylase) can be assumed to be responsible for the higher thermostability of the thermophilic enzyme.

Multiple current reversal in Brownian ratchets.

We address the problem of stationary transport of overdamped Brownian particles in a one-dimensional spatially periodic potential composed of N hills within one period. We show that in a system driven by both thermal equilibrium fluctuations and symmetric dichotomic fluctuations, a proper manipulation of the barrier heights and slopes of the potential leads to multiple drift velocity reversal. Under optimal conditions, the drift velocity as a function of temperature and intensity of dichotomic fluctuations possesses as many as N extrema of alternating signs. There exist N-1 values of a critical temperature which separate regimes of opposite directions of particle transport.

Structural Equilibrium Fluctuations in Mesophilic and Thermophilic α-Amylase

By comparing a mesophilic α-amylase with its thermophilic homolog, we investigated the relationship between thermal stability and internal equilibrium fluctuations. Fourier transform infrared spectroscopy monitoring hydrogen/deuterium (H/D) exchange kinetics and incoherent neutron scattering measuring picosecond dynamics were used to study dynamic features of the folded state at room temperature. Fairly similar rates of slowly exchanging amide protons indicate about the same free energy of stabilization Δ G stab for both enzymes at room temperature. With respect to motions on shorter time scales, the thermophilic enzyme is characterized by an unexpected higher structural flexibility as compared to the mesophilic counterpart. In particular, the picosecond dynamics revealed a higher degree of conformational freedom for the thermophilic α-amylase. The mechanism proposed for increasing thermal stability in the present case is characterized by entropic stabilization and by flattening of the curvature of Δ G stab as a function of temperature.

Microinhomogeneities of glasses of the system PbO–SiO2

Small angle x-ray scattering (SAXS) and Rayleigh–Mandelstam–Brillouin (RMB) light scattering as well as ultrasonic sound velocities have been studied in glasses of the system PbO–SiO2 which has an unusually wide range of glass forming ability. The results of scattering are compared with calculations based on the concept of frozen-in equilibrium thermal fluctuations as the origin of static microinhomogeneities (MIH) in glasses. MIH of compositions seem to be the main source of scattering, and the calculations are found to be in qualitative agreement with both SAXS and RMB measurements. Glasses with PbO content above 40 mol.% are more homogeneous than ideal solutions of PbO and SiO2 whereas MIH in glasses with smaller PbO content are comparable with those expected for ideal solutions of PbO and SiO2. In the latter range SAXS measurements indicate the existence of medium-range order with correlation length of 5–7 Å.

The nature of the hydrated excess proton in water

Explanations for the anomalously high mobility of protons in liquid water began with Grotthuss's idea1, 2 of ‘structural diffusion’ nearly two centuries ago. Subsequent explanations have refined this concept by invoking thermal hopping3, 4, proton tunnelling5, 6 or solvation effects7. More recently, two main structural models have emerged for the hydrated proton. Eigen8, 9 proposed the formation of an H9O4+ complex in which an H3O+ core is strongly hydrogen-bonded to three H2O molecules. Zundel10, 11, meanwhile, supported the notion of an H5O2+ complex in which the proton isshared between two H2O molecules. Here we use ab initio path integral12,13,14 simulations to address this question. These simulations include time-independent equilibrium thermal and quantum fluctuations of all nuclei, and determine interatomic interactions from the electronic structure. We find that the hydrated proton forms a fluxional defect in the hydrogen-bonded network, with both H9O4+ and H5O2+ occurring only in thesense of ‘limiting’ or ‘ideal’ structures. The defect can become delocalized over several hydrogen bonds owing to quantum fluctuations. Solvent polarization induces a small barrier to proton transfer, which is washed out by zero-point motion. The proton can consequently be considered part of a ‘low-barrier hydrogen bond’15, 16, in which tunnelling is negligible and the simplest concepts of transition-state theory do not apply. The rate of proton diffusion is determined by thermally induced hydrogen-bond breaking in the second solvation shell.

Brownian transport controlled by dichotomic and thermal fluctuations

We study transport of Brownian particles in spatially periodic structures, driven by both thermal equilibrium fluctuations and dichotomic noise of zero mean values. Introducing specific scaling, we show that the dimensionless Newton-Langevin type equation governing the motion of Brownian particles is very well approximated by the overdamped dynamics; inertial effects can be neglected because for generic systems dimensionless mass is many orders less than a dimensionless friction coefficient. An exact probability current, proportional to the mean drift velocity of particles, is obtained for a piecewise linear spatially periodic potential. We analyze in detail properties of the macroscopic averaged motion of particles. In dependence on statistics of both sources of fluctuations, the directed transport of particles exhibits such distinctive non-monotonic behavior as: bell-shaped dependence (there exists optimal statistics of fluctuations maximizing velocity) and reversal in the direction of macroscopic motion (there exists critical statistics at which the drift velocity is zero).

Reversible diffusion-controlled reactions in a colloidal kinetic Ising model

The time-dependence of reversible diffusion-controlled reactions in a binary suspension of equal-sized spheres is studied on the basis of the time-correlation matrix of concentration fluctuations in thermal equilibrium. The transient effects are calculated in an Enskog-type approximation from the nonequilibrium distribution function of a pair of diffusing and mutating spheres. The equilibrium radial distribution functions of the system are assumed to be sufficiently well known. The calculation yields predictions of the time-correlation matrix of concentration fluctuations, which can be compared with computer simulations of the model system.

Transient reactions in a binary suspension of diffusing spheres

Transient diffusion-controlled reactions in a binary suspension of equal-sized spheres are studied on the basis of the time-correlation function of concentration fluctuations in thermal equilibrium. The transient effects are characterized by a memory function, which is calculated in an Enskog type approximation from the nonequilibrium distribution function of a pair of diffusing and converting spheres. In Ising model language, the explicit calculation is performed for hard spheres with a square well Ising interaction and with a magnetic field present. The memory function exhibits appreciable dependence on the parameters of the model.

Step capillary waves and equilibrium island shapes on Si(001).

From low-energy electron microscopy measurements of step fluctuations and equilibrium 2D island shapes on Si(001), We determine the temperature dependence of the ${S}_{A}$ and ${S}_{B}$ step edge stiffnesses, free energies, and mobilities. A capillary wave analysis of step stiffnesses is in quantitative agreement with the 2D island shapes, demonstrating the direct relation between equilibrium thermal fluctuations and equilibrium structure. The analysis also suggests that step free energies vanish at \ensuremath{\sim}1200\ifmmode^\circ\else\textdegree\fi{}C, providing direct evidence for a roughening transition. The combined exploration of step dynamics and equilibrium island shapes yields new insight into the basic thermodynamics of Si(001).